Limits at Infinity and Asymptotes

Active learning helps students move beyond procedural calculations to build intuitive understanding of end behavior. When students sketch, compare, and argue about graphs, they connect the algebraic rule to the visual behavior of the function. This kinesthetic and collaborative work makes the abstract concept of limits at infinity concrete and memorable.

• 11th Grade
• Mathematics
• 4 activities
• 15 min–25 min

Learning Objectives

1. Evaluate the limit of a rational function as x approaches positive or negative infinity.
2. Identify the horizontal asymptote of a rational function by analyzing its limit at infinity.
3. Compare the degrees of the numerator and denominator of a rational function to predict its end behavior.
4. Explain how the ratio of leading coefficients determines the value of a limit at infinity when degrees are equal.
5. Calculate the limit of a rational function as x approaches infinity for cases where the numerator's degree is less than the denominator's.

Think-Pair-Share

⏱ 25 min·Small Groups

Three-Case Investigation

Groups receive nine rational functions, three from each degree-comparison case. They compute limits at infinity algebraically, predict horizontal asymptote values, then graph each function to verify. Groups record their findings in a three-column summary and present one example from each case to the class.

Prepare & details

Analyze the connection between limits at infinity and the presence of horizontal asymptotes.

Facilitation Tip: During the Three-Case Investigation, circulate and listen for students comparing the degrees first before attempting algebraic manipulation to reinforce the conceptual foundation.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 25 min·Pairs

Predict-Then-Verify with Desmos

Pairs receive five rational functions and must first predict the horizontal asymptote (or confirm none exists) using the degree comparison rule, then graph the function on Desmos and zoom out to verify end behavior. They annotate each graph with the computed limit and note any surprises.

Prepare & details

Predict the end behavior of rational functions using limits at infinity.

Facilitation Tip: While students use Desmos in Predict-Then-Verify, ask them to zoom out slowly so they observe how the graph behaves as x grows, not just the initial view.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: What Happens as x Gets Very Large?

Partners build a numerical table for a rational function at x equals 10, 100, 1000, and 10000, then predict the limit at infinity numerically before confirming algebraically. Discussion centers on whether the function ever actually reaches its horizontal asymptote.

Prepare & details

Explain how limits at infinity describe the long-term behavior of a function.

Facilitation Tip: For the Think-Pair-Share, provide a mix of simple and complex examples so students confront the crossing behavior directly rather than assuming all rational functions stay on one side.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Gallery Walk

⏱ 20 min·Whole Class

Gallery Walk: Match the Function to the Asymptote

Posters around the room each display a rational function. Students visit each poster and write the horizontal asymptote (or none) and the limit as x approaches infinity on a sticky note. A final comparison wall at the end of the gallery lets students resolve disagreements as a class.

Prepare & details

Analyze the connection between limits at infinity and the presence of horizontal asymptotes.

Facilitation Tip: During the Gallery Walk, place the function cards at eye level and ask students to annotate their reasoning on sticky notes so their thinking is visible to others.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Start with graphical intuition before formal rules. Students benefit from seeing multiple examples where the graph crosses the asymptote or behaves unexpectedly, which counters early misconceptions. Emphasize that horizontal asymptotes describe end behavior only, not the entire graph. Use consistent language like 'the limit is L' rather than 'the function approaches L' to prepare students for formal limit notation.

What to Expect

Students will confidently connect polynomial degrees to horizontal asymptotes and explain why functions cross or approach these asymptotes. They will use precise language to describe limits at infinity, distinguishing between finite limits and infinite growth. Small-group discussions and graphical explorations will reveal their reasoning.

Watch Out for These Misconceptions

Common Misconception

During the Think-Pair-Share activity, watch for students claiming a function can never cross its horizontal asymptote.

What to Teach Instead

Provide a function like f(x) = (x)/(x^2 + 1) and ask students to find a value of x where f(x) = 0.25, which is above the asymptote y = 0, to directly confront the misconception using their own calculations.

Common Misconception

During the Three-Case Investigation, listen for students saying the limit does not exist when the numerator's degree is larger.

What to Teach Instead

Use the activity’s case cards to have students write the limit as 'limit is infinity' and explain why this is still informative end behavior, not a lack of limit.

Assessment Ideas

Quick Check

After the Three-Case Investigation, provide a handout with 3-4 rational functions and ask students to calculate the limit as x approaches infinity for each, stating if a horizontal asymptote exists and its equation.

Discussion Prompt

During the Think-Pair-Share, listen for students explaining why the limit of f(x) = (x^2 + 2x)/(x^2 - 4) as x approaches infinity is 1, referencing the degrees of the polynomials.

Exit Ticket

After the Gallery Walk, ask students to write a rational function with a horizontal asymptote at y = 3 and explain in one sentence how they determined this based on the function's structure.

Extensions & Scaffolding

• Challenge: Ask students to construct a rational function that crosses its horizontal asymptote three times.
• Scaffolding: Provide partially completed tables for comparing degrees and their effects on limits at infinity.
• Deeper exploration: Have students research oblique asymptotes and explain how polynomial long division reveals these slanted lines.

Key Vocabulary

Limit at Infinity	Describes the behavior of a function as the input variable x increases or decreases without bound, approaching positive or negative infinity.
Horizontal Asymptote	A horizontal line that the graph of a function approaches as x approaches positive or negative infinity. It represents the function's value at these extremes.
End Behavior	The behavior of a function's output values as the input values become very large (positive or negative).
Rational Function	A function that can be expressed as the ratio of two polynomial functions, where the denominator is not the zero polynomial.